Floating-point Research Articles

A novel MissForest-based missing values imputation approach with recursive feature elimination in medical applications

BackgroundMissing values in datasets present significant challenges for data analysis, particularly in the medical field where data accuracy is crucial for patient diagnosis and treatment. Although MissForest (MF) has demonstrated efficacy in imputation research and recursive feature elimination (RFE) has proven effective in feature selection, the potential for enhancing MF through RFE integration remains unexplored.MethodsThis study introduces a novel imputation method, “recursive feature elimination-MissForest” (RFE-MF), designed to enhance imputation quality by reducing the impact of irrelevant features. A comparative analysis is conducted between RFE-MF and four classical imputation methods: mean/mode, k-nearest neighbors (kNN), multiple imputation by chained equations (MICE), and MF. The comparison is carried out across ten medical datasets containing both numerical and mixed data types. Different missing data rates, ranging from 10 to 50%, are evaluated under the missing completely at random (MCAR) mechanism. The performance of each method is assessed using two evaluation metrics: normalized root mean squared error (NRMSE) and predictive fidelity criterion (PFC). Additionally, paired samples t-tests are employed to analyze the statistical significance of differences among the outcomes.ResultsThe findings indicate that RFE-MF demonstrates superior performance across the majority of datasets when compared to four classical imputation methods (mean/mode, kNN, MICE, and MF). Notably, RFE-MF consistently outperforms the original MF, irrespective of variable type (numerical or categorical). Mean/mode imputation exhibits consistent performance across various scenarios. Conversely, the efficacy of kNN imputation fluctuates in relation to varying missing data rates.ConclusionThis study demonstrates that RFE-MF holds promise as an effective imputation method for medical datasets, providing a novel approach to addressing missing data challenges in medical applications.

The influence of parasitic modes on stable lattice Boltzmann schemes and weakly unstable multi-step Finite Difference schemes

Numerical analysis for linear constant-coefficient multi-step Finite Difference schemes is a longstanding topic, developed approximately fifty years ago. It relies on the stability of the scheme, and thus—within the L2 setting—on the absence of multiple roots of the amplification polynomial on the unit circle. This allows for the decoupling, while discussing the convergence of the method, of the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that the methods can be essentially studied as if they had only one step. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we demonstrate that in the case of “weakly” unstable Finite Difference schemes with multiple roots on the unit circle, although the schemes may remain stable, considering parasitic modes is essential in studying their consistency and, consequently, their convergence. This research was prompted by unexpected numerical results on stable lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Unlike Finite Difference schemes, rigorous numerical analysis for lattice Boltzmann schemes is a contemporary topic with much left for future discoveries. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of fourth-order schemes. However, this assumption proved incorrect for weakly unstable Finite Difference schemes and for stable lattice Boltzmann methods. This borderline scenario underscores that particular care must be adopted for lattice Boltzmann schemes, and the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors concerning Finite Difference schemes. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle nonlinear systems of conservation laws relying on a Jin-Xin approximation and high-order splitting formulæ.

Fighting against forest fire: A lightweight real-time detection approach for forest fire based on synthetic images

Forest fires are known for their high level of randomness and unpredictability, which often lead to significant ecological damage and human life loss. Existing forest fire detection technologies are not capable of detecting small-scale flames or smoke in real time, thus failing to meet the demands of real-time detection of forest fires using Unmanned Aerial Vehicles (UAVs). To overcome these limitations, we propose an efficient and lightweight forest fire detection method that utilizes synthetic images and UAVs to achieve real-time and high-precision detection of forest fires against complex backgrounds. Firstly, we propose the Dilation Repconv Cross Stage Partial Network (DRCSPNet), which enhances the detection capabilities for multiscale flames and smoke using multi-branch parallel joint dilation convolution and batch normalization, while effectively extracting features from different stages of forest fires. Secondly, to mitigate challenges associated with extreme lighting in forest scenes and large contrast variation in fire images, we propose a Global Mixed-Attention (GMA) model across feature pyramids to enhance information lost in high-dimensional feature maps and increase the robustness of the model through a multiscale fusion strategy. Finally, we present the Lite-Path Aggregation Network (Lite-PAN) with varying scales to improve effective feature flow for multilevel forest fires, addressing challenges that arise from various climatic conditions. Furthermore, we employ Unreal Engine 5 to generate forest fire datasets in four scenarios to address the issue of relatively limited aerial forest fire datasets. According to the results of the experiment, our proposed method achieves 58.39% mAP(mean Average Precision) with 5.703 GFLOPs (Giga Floating Point Operations Per Second) while yielding a frame rate of 33.5 Frames Per Second (FPS) on NVIDIA Jetson NX. Extensive experiment results demonstrate our method has the advantage of being in real time, extremely accurate, and easily implementable compared to state-of-the-art techniques.

Performance Evaluation of FPGA-Based Design of Modified Chua Oscillator

The chaotic systems are one of the most important areas that increasing the popularity and are actively used in several fields. One of the most essential structures in chaotic systems is chaotic oscillator which generates chaotic signals. IQ-Math and floating point number systems are one of the preferred number standards. Presented in this study, the Modified Chua chaotic oscillator has been designed to work on FPGA chips using fixed point and floating point number systems. Euler numeric algorithm has been used for the design of the Modified Chua chaotic oscillator. The first section of the study, Modified Chua chaotic system based on fixed point has been composed the model in the Matlab Simulink and transformed to VHDL with the help of Matlab HDL Coder Toolbox. The second section of the study, Modified Chua chaotic oscillator has been designed with VHDL based on floating point. Modified Chua chaotic oscillators which are composed with two different number standards have been tested using Xilinx ISE Design Tools in VHDL. Modified Chua chaotic oscillators which have two different numbers standards and designed, are synthesized for Virtex-6 on ML605 FPGA development board with Xilinx ISE Design Tools 14.2 program. Values that are achieved from the process of synthesizing and maximum operating frequency have been presented. As a result, the study has obtained that fixed point number standard maximum operating frequency is 50.242 MHz and floating point number standard maximum operating frequency is 273.631 MHz.

Mixed-precision pre-pivoting strategy for the LU factorization

This paper investigates the efficient application of half-precision floating-point (FP16) arithmetic on GPUs for boosting LU decompositions in double (FP64) precision. Addressing the motivation to enhance computational efficiency, we introduce two novel algorithms: Pre-Pivoted LU (PRP) and Mixed-precision Panel Factorization (MPF). Deployed in both hybrid CPU-GPU setups and native GPU-only configurations, PRP identifies pivot lists through LU decomposition computed in reduced precision and subsequently reorders matrix rows in FP64 precision before executing LU decomposition without pivoting. Two variants of PRP, namely hPRP and xPRP, are introduced, differing in their computation of pivot lists in full half-precision or mixed half-single precision. The MPF algorithm generates FP64 LU factorization while internally utilizing hPRP for panel factorization, showcasing accuracy on par with standard DGETRF but with superior speed. The study further explores auxiliary functions required for the native mode implementation of PRP variants and MPF.

Lightweight skin cancer detection IP hardware implementation using cycle expansion and optimal computation arrays methods

Skin cancer is recognized as one of the most perilous diseases globally. In the field of medical image classification, precise identification of early-stage skin lesions is imperative for accurate diagnosis. However, deploying these algorithms on low-cost devices and attaining high-efficiency operation with minimal energy consumption poses a formidable challenge due to their intricate computational demands. This study proposes a lightweight hardware design based on a convolutional neural network (CNN) for real-time processing of skin disease classifiers on portable devices. Our skin cancer recognition processor utilizes an optimally parallel designed processing engine (PE) for global computation, which greatly reduces hardware resource utilization by multiplexing of computational unit circuits. In addition, a design approach that provides loop unrolling effectively reduces the number of data accesses, thereby reducing computational complexity and logic resource requirements. The hardware circuits in this design perform data inference in convolutional, pooling, and fully connected layers based on 16-bit floating-point numbers. Evaluation of the HAM10000 database dataset shows that the architecture achieves an average classification accuracy of 97.8 %. We are the first to implement an all-hardware FPGA-based skin cancer detection platform that offers a 3.5x speedup in recognition compared to existing skin cancer accelerators at 50 MHz while consuming only 0.48 W of power. The implementation of this hardware architecture meets the major constraints of portable devices, featuring low resource utilization, low power consumption, and cost-effectiveness, while still providing efficient classification and high accuracy results.

FPCC: Detecting Floating-Point Errors via Chain Conditions

Floating-point arithmetic is notorious for its rounding errors, which can propagate and accumulate, leading to unacceptable results. Detecting inputs that can trigger significant floating-point errors is crucial for enhancing the reliability of numerical programs. Existing methods for generating error-triggering inputs often rely on costly shadow executions that involve high-precision computations or suffer from false positives. This paper introduces chain conditions to capture the propagation and accumulation of floating-point errors, using them to guide the search for error-triggering inputs. We have implemented a tool named FPCC and evaluated it on 88 functions from the GNU Scientific Library, as well as 21 functions with multiple inputs from previous research. The experimental results demonstrate the effectiveness and efficiency of our approach: (1) FPCC achieves 100% accuracy in detecting significant errors for the reported rank-1 inputs, while 72.69% rank-1 inputs from the state-of-the-art tool ATOMU can trigger significant errors. Overall, 99.64% (1049/1053) of the inputs reported by FPCC can trigger significant errors, whereas only 19.45% (141/723) of the inputs reported by ATOMU can trigger significant errors; (2) FPCC exhibits a 2.17x speedup over ATOMU in detecting significant errors; (3) FPCC also excels in supporting functions with multiple inputs, outperforming the state-of-the-art technique. To facilitate further research in the community, we have made FPCC available on GitHub at https://github.com/DataReportRe/FPCC.

Sharkovskii’s theorem and the limits of digital computers for the simulation of chaotic dynamical systems

Chaos is a unique paradigm in classical physics within which systems exhibit extreme sensitivity to the initial conditions. Thus, they need to be handled using probabilistic methods commonly based on ensembles. However, initial conditions generated by digital computers fall within the sparse set of discrete IEEE floating point numbers which have non-uniform distributions along the real axis. Therefore, there are many missing initial conditions whose absence might be expected to degrade the computed statistical properties of chaotic systems. The universality of this problem is enshrined in Sharkovskii’s theorem which is the simplest mathematical statement of the fact that no finite number representation of a chaotic dynamical system can account for all of its properties and shows that the precision of the representation limits the accuracy of the resulting digital behaviour.

Effect of bit-size reduced half-precision floating-point format on image pixel characterization for AI applications

Image processing is an essential first step towards fully utilizing robotics, deep learning, and machine learning techniques. Using techniques like image enhancement, restoration, and segmentation are able to extract pertinent information from images and use it for task execution and decision-making. However, the hardware implementation of these algorithms demands more delay, area, and power. This work proposes a new mantissa bit-size reduced half-precision floating-point format for processing and characterizing image pixels for machine learning algorithms. In the realm of imaging, lowering the mantissa bit-size in floating-point can conserve area and power when utilized for internal calculations. Together with the area-power reduction, there is also a progressive reduction in image quality. For any image application, it becomes necessary to monitor the area and power trade-offs related to the amount of bits used to process the raw data. This work assists in selecting the bit-size for internal computations based on the accuracy requirements of the application by reporting the area, and power for various bit size reductions. The updated pixel values after applying the mantissa bit-size reduction are displayed in this study, along with a theoretical explanation of its inaccuracy. Since multipliers and adders are needed for the majority of mathematical equations in machine learning image algorithms, they are developed later in this work to process the image. The processed image is based on various adjusted pixel values, and the experimental findings demonstrate 75.2% to 21.3% area optimization, and 66.43% to 20.4% power optimization. However, determining the PSNR and MSE values of the processed image allows for quality validation.

Information hiding using approximate POSIT representation

AbstractPOSIT is a numerical system consists of sign, regime, exponential, and fraction bits to overcome some limitations of the IEEE‐754 floating point (FP) representation. This work proposes a technique to hide critical information in the least significant bits (LSBs) of the POSIT FP representation by exploiting approximate computing (AC). The proposed technique, called information hiding using POSIT and LSB (IHUPL), explores the opportunities offered by both the POSIT representation and the AC to hide information with a minimum loss in accuracy and other performance metrics. IHUPL offers two options for hiding information: either by embedding one digit of each character into each pixel or by embedding all the digits of the character at the same time into each pixel of the alpha‐red, green, blue, or black/white image. Experimental results are evaluated for benchmark images and showed that IHUPL enhanced the accuracy of embedding data into LSB of POSIT FP by an average of 5%, the image quality improvement rates of IHUPL are 19% and 16% for options 1 and 2, respectively. Besides the encoding method using IHUPL, the paper outlines an extraction decoding technique that saves the original replacement bits in a key‐image to recover the hidden security message.

FPGA‐Based Implementation of Real‐Time Cardiologist‐Level Arrhythmia Detection and Classification in Electrocardiograms Using Novel Deep Learning

ABSTRACTCardiac arrhythmia refers to irregular heartbeats caused by anomalies in electrical transmission in the heart muscle, and it is an important threat to cardiovascular health. Conventional monitoring and diagnosis still depend on the laborious visual examination of electrocardiogram (ECG) devices, even though ECG signals are dynamic and complex. This paper discusses the need for an automated system to assist clinicians in efficiently recognizing arrhythmias. The existing machine‐learning (ML) algorithms have extensive training cycles and require manual feature selection; to eliminate this, we present a novel deep learning (DL) architecture. Our research introduces a novel approach to ECG classification by combining the vision transformer (ViT) and the capsule network (CapsNet) into a hybrid model named ViT‐Cap. We conduct necessary preprocessing operations, including noise removal and signal‐to‐image conversion using short‐time Fourier transform (SIFT) and continuous wavelet transform (CWT) algorithms, on both normal and abnormal ECG data obtained from the MIT‐BIH database. The proposed model intelligently focuses on crucial features by leveraging global and local attention to explore spectrogram and scalogram image data. Initially, the model divides the images into smaller patches and linearly embeds each patch. Features are then extracted using a transformer encoder, followed by classification using the capsule module with feature vectors from the ViT module. Comparisons with existing conventional models show that our proposed model outperforms the original ViT and CapsNet in terms of classification accuracy for both binary and multi‐class ECG classification. The experimental findings demonstrate an accuracy of 99% on both scalogram and spectrogram images. Comparative analysis with state‐of‐the‐art methodologies confirms the superiority of our framework. Additionally, we configure a field‐programmable gate array (FPGA) to implement the proposed model for real‐time arrhythmia classification, aiming to enhance user‐friendliness and speed. Despite numerous suggestions for high‐performance FPGA accelerators in the literature, our FPGA‐based accelerator utilizes optimization of loop parallelization, FP data, and multiply accumulation (MAC) unit. Our accelerator architecture achieves a 57% reduction in processing time and utilizes fewer resources compared to a floating‐point (FlP) design.

Research on the Behavior Recognition of Beef Cattle Based on the Improved Lightweight CBR-YOLO Model Based on YOLOv8 in Multi-Scene Weather.

In modern animal husbandry, intelligent digital farming has become the key to improve production efficiency. This paper introduces a model based on improved YOLOv8, Cattle Behavior Recognition-YOLO (CBR-YOLO), which aims to accurately identify the behavior of cattle. We not only generate a variety of weather conditions, but also introduce multi-target detection technology to achieve comprehensive monitoring of cattle and their status. We introduce Inner-MPDIoU Loss and we have innovatively designed the Multi-Convolutional Focused Pyramid module to explore and learn in depth the detailed features of cattle in different states. Meanwhile, the Lightweight Multi-Scale Feature Fusion Detection Head module is proposed to take advantage of deep convolution, achieving a lightweight network architecture and effectively reducing redundant information. Experimental results prove that our method achieves an average accuracy of 90.2% with a reduction of 3.9 G floating-point numbers, an increase of 7.4%, significantly better than 12 kinds of SOTA object detection models. By deploying our approach on monitoring computers on farms, we expect to advance the development of automated cattle monitoring systems to improve animal welfare and farm management.

MicroFloatingPoints.jl: providing very small IEEE 754-compliant floating-point types

MicroFloatingPoints.jl: providing very small IEEE 754-compliant floating-point types

Floating-point photonic iterative solver demonstrated for Newton–Raphson method

Though photonic computing systems offer advantages in speed, scalability, and power consumption, they often have a limited dynamic encoding range due to low signal-to-noise ratios. Compared to digital floating-point encoding, photonic fixed-point encoding limits the precision of photonic computing when applied to scientific problems. In the case of iterative algorithms such as those commonly applied in machine learning or differential equation solvers, techniques like precision decomposition and residue iteration can be applied to increase accuracy at a greater computing cost. However, the analog nature of photonic symbols allows for modulation of both amplitude and frequency, opening the possibility of encoding both the significand and exponent of floating-point values on photonic computing systems to expand the dynamic range without expending additional energy. With appropriate schema, element-wise floating-point multiplication can be performed intrinsically through the interference of light. Herein, we present a method for configurable, signed, floating-point encoding and multiplication on a limited precision photonic primitive consisting of a directly modulated Mach–Zehnder interferometer. We demonstrate this method using Newton's method to find the Golden Ratio within ±0.11%, with six-level exponent encoding for a signed trinary digit-equivalent significand, corresponding to an effective increase of 243× in the photonic primitive's dynamic range.

AMP: Total Variation Reduction for Lossless Compression via Approximate Median-based Preconditioning

With the increasing scale of cloud computing applications of next-generation embedded systems, a major challenge that domain scientists are facing is how to efficiently store and analyze the vast volume of output data. Compression can reduce the amount of data that needs to be transferred and stored. However, most of the large datasets are in floating-point format, which exhibits high entropy. As a result, existing lossless compressors cannot provide enough performance for such applications. To address this problem, we propose a total variation reduction method for improving the compression ratio of lossless compressors (namely, FPC + and FPZIP + ), which employs a median-based hyperplane to precondition the data. In particular, we first try to exploit the space-filling curve (SFC), a well-known technique to preserve data locality for a multi-dimensional dataset. We show and explain why a raw SFC, such as Hilbert and Z-order curves, cannot improve the compression ratio. Then, we explore the opportunity and theoretical feasibility of the proposed total variation reduction-based algorithm. The experiment results show the effectiveness of the proposed method. The compression ratios are improved up to 48.2% (20.6% on average) for FPZIP and 42.4% (18.4% on average) for FPC. Moreover, through observing the time composition of the proposed method, it is found that the median finding holds a high percentage of the execution time. Hence, we further introduce an approximate median finding algorithm, providing a linear-time overhead reduction scheme. The experiment results clearly demonstrate that this algorithm reduces execution time by an average of 56.7% and 40.7% compared to FPC + and FPZIP + , respectively.

Ranking of 10 Global One-Arc-Second DEMs Reveals Limitations in Terrain Morphology Representation

At least 10 global digital elevation models (DEMs) at one-arc-second resolution now cover Earth. Comparing derived grids, like slope or curvature, preserves surface spatial relationships, and can be more important than just elevation values. Such comparisons provide more nuanced DEM rankings than just elevation root mean square error (RMSE) for a small number of points. We present three new comparison categories: fraction of unexplained variance (FUV) for grids with continuous floating point values; accuracy metrics for integer code raster classifications; and comparison of stream channel vector networks. We compare six global DEMs that are digital surface models (DSMs), and four edited versions that use machine learning/artificial intelligence techniques to create a bare-earth digital terrain model (DTM) for different elevation ranges: full Earth elevations, under 120 m, under 80 m, and under 10 m. We find edited DTMs improve on elevation values, but because they do not incorporate other metrics in their training they do not improve overall on the source Copernicus DSM. We also rank 17 common geomorphic-derived grids for sensitivity to DEM quality, and document how landscape characteristics, especially slope, affect the results. None of the DEMs perform well in areas with low average slope compared to reference DTMs aggregated from 1 m airborne lidar data. This indicates that accurate work in low-relief areas grappling with global climate change should use airborne lidar or very high resolution image-derived DTMs.

Does combining numerous data types in multi-omics data improve or hinder performance in survival prediction? Insights from a large-scale benchmark study

BackgroundPredictive modeling based on multi-omics data, which incorporates several types of omics data for the same patients, has shown potential to outperform single-omics predictive modeling. Most research in this domain focuses on incorporating numerous data types, despite the complexity and cost of acquiring them. The prevailing assumption is that increasing the number of data types necessarily improves predictive performance. However, the integration of less informative or redundant data types could potentially hinder this performance. Therefore, identifying the most effective combinations of omics data types that enhance predictive performance is critical for cost-effective and accurate predictions.MethodsIn this study, we systematically evaluated the predictive performance of all 31 possible combinations including at least one of five genomic data types (mRNA, miRNA, methylation, DNAseq, and copy number variation) using 14 cancer datasets with right-censored survival outcomes, publicly available from the TCGA database. We employed various prediction methods and up-weighted clinical data in every model to leverage their predictive importance. Harrell’s C-index and the integrated Brier Score were used as performance measures. To assess the robustness of our findings, we performed a bootstrap analysis at the level of the included datasets. Statistical testing was conducted for key results, limiting the number of tests to ensure a low risk of false positives.ResultsContrary to expectations, we found that using only mRNA data or a combination of mRNA and miRNA data was sufficient for most cancer types. For some cancer types, the additional inclusion of methylation data led to improved prediction results. Far from enhancing performance, the introduction of more data types most often resulted in a decline in performance, which varied between the two performance measures.ConclusionsOur findings challenge the prevailing notion that combining multiple omics data types in multi-omics survival prediction improves predictive performance. Thus, the widespread approach in multi-omics prediction of incorporating as many data types as possible should be reconsidered to avoid suboptimal prediction results and unnecessary expenditure.

Profiling of collagen and extracellular matrix deposition from cell culture using in vitro ExtraCellular matrix mass spectrometry imaging (ivECM-MSI)

While numerous approaches have been reported towards understanding single cell regulation, there is limited understanding of single cell production of extracellular matrix phenotypes. Collagens are major proteins of the extracellular microenvironment extensively used in basic cell culture, tissue engineering, and biomedical applications. However, identifying compositional regulation of collagen remains challenging. Here, we report the development of In vitro ExtraCellular Matrix Mass Spectrometry Imaging (ivECM-MSI) as a tool to rapidly and simultaneously define collagen subtypes from coatings and basic cell culture applications. The tool uses the mass spectrometry imaging platform with reference libraries to produce visual and numerical data types. The method is highly integrated with basic in vitro strategies as it may be used with conventional cell chambers on minimal numbers of cells and with minimal changes to biological experiments. Applications tested include semi-quantitation of collagen composition in culture coatings, time course collagen deposition, deposition altered by gene knockout, and changes induced by drug treatment. This approach provides new access to proteomic information on how cell types respond to and change the extracellular microenvironment and provides a holistic understanding of both the cell and extracellular response.

Improving protection of falcon electronic signature software implementations against attacks based on floating point noise

The object of this study is digital signatures. The Falcon digital signature scheme is one of the finalists in the NIST post-quantum cryptography competition. Its distinctive feature is the use of floating-point arithmetic, which leads to the possibility of a key recovery attack with two non-matching signatures formed under special conditions. The work considers the task to improve the Falcon in order to prevent such attacks, as well as the use of fixed-point calculations instead of floating-point calculations in the Falcon scheme. The main results of the work are proposals for methods on improving Falcon's security against attacks based on the use of floating-point calculations. These methods for improving security differ from others in the use of fixed-point calculations with specific experimentally determined orders of magnitude in one case and proposals for modifying procedures during the execution of which the conditions for performing an attack on implementation level arise in the second case. As a result of the analysis, the probability of a successful attack on the recovery of the secret key for the reference implementation of the Falcon was clarified. Specific places in the code that make the attack possible have been localized and code modifications have been suggested that make the attack impossible. In addition, the necessary scale for fixed-point calculations was determined, at which it is possible to completely get rid of floating-point calculations. The results could be used to qualitatively improve the security of existing digital signatures. This will make it possible to design more reliable and secure information systems using digital signatures. In addition, the results could be implemented in existing systems to ensure their resistance to modern threats

DESIGN AND PERFORMANCE ANALYSIS OF TERNARY LOGIC BASED ALU USING DOUBLE PRECISION FLOATING POINT

DESIGN AND PERFORMANCE ANALYSIS OF TERNARY LOGIC BASED ALU USING DOUBLE PRECISION FLOATING POINT